Pith Number

pith:RPVPFU2J

pith:2026:RPVPFU2JO7B23WCUNHIUIXCXTC

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Scalable Bi-causal Optimal Transport via KL Relaxation and Policy Gradients

Haoyang Cao, Jesse Hoekstra, Renyuan Xu, Ruixun Zhang, Yumin Xu

KL relaxation turns bi-causal optimal transport into a policy-gradient problem

arxiv:2605.17271 v1 · 2026-05-17 · math.OC · cs.LG

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Claims

C1strongest claim

We establish dynamic programming principles for both the original and relaxed formulations, prove that the relaxed problem converges to the original bi-causal OT problem as the penalty grows, and derive explicit policy-gradient representations for the relaxed objective. Building on these results, we propose a practical policy-gradient algorithm with unbiased mini-batch estimators, variance reduction, and nonasymptotic regret guarantees.

C2weakest assumption

The framework assumes that the KL relaxation preserves the recursive structure of the bi-causal problem sufficiently for dynamic programming and policy gradient methods to apply directly, and that marginal laws can be sampled to enable the stochastic optimization procedure described.

C3one line summary

A KL-relaxed formulation of bi-causal optimal transport is solved via policy gradients with proven convergence to the original problem and nonasymptotic regret guarantees for the resulting algorithm.

References

23 extracted · 23 resolved · 0 Pith anchors

[1] Here, the first inequality follows fromJπ n+1 ≥V n+1 and the monotonicity of operatorQπ n+1, and the last inequality follows from (15)

[2] NX n=0 cn(Yn, Y ′ n) # =E π

[3] 43 The definition ofWbc in (5) as the infimum overπ∈ΠΠΠbc yieldsW bc(µ, µ′)≥inf π0∈Π(µ0,µ′ 0) Eπ0[V0]

[4] Notice that (A.3) holds for any initial coupling inΠ(µ0, µ′ 0)

[5] To establish the second equivalent representation, notice thatγ ϵ ∈ M bc(µ, µ′)⊂Π ΠΠbc and Wbc(µ, µ′) = inf π0∈Π(µ0,µ′

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Receipt and verification

First computed	2026-05-20T00:03:49.075129Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8beaf2d34977c3add85469d1445c579894158e37994a26423efae65a99af3cf3

Aliases

arxiv: 2605.17271 · arxiv_version: 2605.17271v1 · doi: 10.48550/arxiv.2605.17271 · pith_short_12: RPVPFU2JO7B2 · pith_short_16: RPVPFU2JO7B23WCU · pith_short_8: RPVPFU2J

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/RPVPFU2JO7B23WCUNHIUIXCXTC \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 8beaf2d34977c3add85469d1445c579894158e37994a26423efae65a99af3cf3

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "34079c9295c1885dc93323d765ffdc717e05761879e479d3b2d231944c3c02d6",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-17T05:41:01Z",
    "title_canon_sha256": "6af38eec6ac56c9daa9e2771420cfc020187e38bd974263dee9327ac3819c828"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17271",
    "kind": "arxiv",
    "version": 1
  }
}